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cos^2*x+sin(x)+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 cos (x) + sin(x) + 1 = 0
$$\left(\sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 1 = 0$$
cambiamos
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} + 1 = 0$$
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 2 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = 1$$
$$c = 2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = -1$$
$$w_{2} = 2$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$\left(\sin{\left(x \right)} + \cos^{2}{\left(x \right)}\right) + 1 = 0$$
cambiamos
$$\sin{\left(x \right)} + \cos^{2}{\left(x \right)} + 1 = 0$$
$$- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 2 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
a*w^2 + b*w + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -1$$
$$b = 1$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-1) * (2) = 9
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = -1$$
$$w_{2} = 2$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
Respuesta rápida
[src]
-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x2 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x3 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
\ \2 2 // \ \2 2 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
x3 = 2*re(atan(1/2 + sqrt(3)*i/2)) + 2*i*im(atan(1/2 + sqrt(3)*i/2))
Suma y producto de raíces
[src]
suma
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ pi | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || - -- + 2*re|atan|- - -------|| + 2*I*im|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- + -------|| 2 \ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\left(- \frac{\pi}{2} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 || pi | |1 I*\/ 3 || | |1 I*\/ 3 ||
2*re|atan|- + -------|| + 2*re|atan|- - -------|| - -- + 2*I*im|atan|- + -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 // 2 \ \2 2 // \ \2 2 //
$$- \frac{\pi}{2} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
producto
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -pi | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| ----*|2*re|atan|- - -------|| + 2*I*im|atan|- - -------|||*|2*re|atan|- + -------|| + 2*I*im|atan|- + -------||| 2 \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$- \frac{\pi}{2} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 |||
-2*pi*|I*im|atan|- + -------|| + re|atan|- + -------|||*|I*im|atan|- - -------|| + re|atan|- - -------|||
\ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$- 2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
-2*pi*(i*im(atan(1/2 + i*sqrt(3)/2)) + re(atan(1/2 + i*sqrt(3)/2)))*(i*im(atan(1/2 - i*sqrt(3)/2)) + re(atan(1/2 - i*sqrt(3)/2)))
Respuesta numérica
[src]
x1 = 48.6946859158554
x2 = -14.1371668385321
x3 = 61.2610570434674
x4 = 54.9778711510713
x5 = 61.2610564670687
x6 = 42.4115014315384
x7 = -32.9867231539387
x8 = -39.2699081135452
x9 = -89.5353911652882
x10 = -64.402649255422
x11 = -76.9690197221984
x12 = 98.9601688588295
x13 = 10.9955745697675
x14 = -89.5353907479368
x15 = -76.969020305975
x16 = -1.57079643043874
x17 = -51.8362786896082
x18 = 4.7123887595313
x19 = 92.6769830723585
x20 = 86.393797887839
x21 = 80.1106131412134
x22 = -45.5530935892791
x23 = 67.5442420465931
x24 = -7.85398149791991
x25 = -102.101761883247
x26 = -20.4203520230544
x27 = -70.6858349653034
x28 = -26.7035372612266
x29 = -83.2522052322402
x30 = -70.6858344139979
x31 = 17.2787598912651
x32 = 42.4115007285955
x33 = 98.9601683040195
x34 = -20.4203521614409
x35 = 54.9778717146949
x36 = 36.1283156889147
x37 = 48.6946861099035
x38 = 193.207948237308
x39 = -26.7035378220867
x40 = 29.8451303212917
x41 = -39.2699083966096
x42 = 10.9955739984145
x43 = -32.9867225758835
x44 = -83.2522055525629
x45 = 23.5619449395086
x46 = 73.8274274807538
x47 = 67.5442422891227
x48 = 73.8274268696112
x49 = 92.6769832132472
x50 = 4.71238900072427
x51 = -58.1194639987802
x52 = -64.4026491794358
x53 = 17.2787593220659
x54 = 1858.252054715
x55 = -95.8185758681024
x56 = -7824.136503369
x57 = 23.5619451331326
x58 = 48.6946862428019
x58 = 48.6946862428019